Editing Pseudo-differential operator (section)

==Properties==
Linear differential operators of order m with smooth bounded coefficients are pseudo-differential
operators of order ''m''.
The composition ''PQ'' of two pseudo-differential operators ''P'',&nbsp;''Q'' is again a pseudo-differential operator and the symbol of ''PQ'' can be calculated by using the symbols of ''P'' and ''Q''. The adjoint and transpose of a pseudo-differential operator is a pseudo-differential operator.

If a differential operator of order ''m'' is [[elliptic differential operator|(uniformly) elliptic]] (of order ''m'')
and invertible, then its inverse is a pseudo-differential operator of order &minus;''m'', and its symbol can be calculated. This means that one can solve linear elliptic differential equations more or less explicitly
by using the theory of pseudo-differential operators.

Differential operators are ''local'' in the sense that one only needs the value of a function in a neighbourhood of a point to determine the effect of the operator. Pseudo-differential operators are ''pseudo-local'', which means informally that when applied to a [[Schwartz distribution|distribution]] they do not create a singularity at points where the distribution was already smooth.

Just as a differential operator can be expressed in terms of ''D''&nbsp;=&nbsp;&minus;id/d''x'' in the form

:<math>p(x, D)\,</math>

for a [[polynomial]] ''p'' in ''D'' (which is called the ''symbol''), a pseudo-differential operator has a symbol in a more general class of functions. Often one can reduce a problem in analysis of pseudo-differential operators to a sequence of algebraic problems involving their symbols, and this is the essence of [[microlocal analysis]].